Every day, I am faced with the dilemma of explaining some complex phenomena or the other to my students. In a given time that I spend with my students in the class, how do I make them understand a difficult topic? To realize my goal, I tell “lies to students”, a phrase I coined after reading fantasy writer Terry Prachett’s “lies to children”. A lie, according to Prachett, is “a statement that is false, but which nevertheless leads the child’s mind towards a more accurate explanation”. We use “lies” as tools if we aim to be effective teachers in conveying a complex subject in an understandable language. Like all teachers, I am honest too. Therefore, I actually say something like this: “Look, I am going to tell this lie to make you understand better”. Tomorrow if my students have to shape into good engineers or scientists, they need to master this “art of telling lies” because everything in science is an approximation to reality or “a lie”. If you disagree with what I am saying, perhaps you would find comfort in what Richard Levins said, “Our truth is the intersection of independent lies”. He further says “Even the most flexible models have artificial assumptions. There is always room for doubt as to whether a result depends on the essentials of a model or on the details of the simplifying assumptions.”
Approximations are an essential part of scientific pursuits. When I use approximations to explain a concept, my students are often perplexed. They think I am trivializing a profound concept. After making approximations, often unrealistic or unphysical, a lengthy and complex equation reduces to a simple form and gives us a greater insight about the working of a system. I could then see a smile on the faces of the students. Bertrand Russell said in “The Scientific Outlook” (1931), “Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man.”
We habitually use approximations during our lectures. We often do not, however, emphatically underline the fact that many of the laws or theories we study are actually mere approximations. We tend to pretend during lectures that the laws we are presenting are truths. This is because, as humans, we all suffer from a syndrome called confirmation bias which forces us to seek ideas that fit our current views than critically think and contradict with what we hold as truth. Take, for example, Ohm’s law or Henry’s law. How often did we tell our students that they are in fact not “laws” at all and that they are simply approximations to experimental observations. When a student asked me to add my perspective on approximations, I almost withdrew. I did not know how to approach this subject of approximations which seemed too complex to me. Approximations do hold the key to the evolution of modern knowledge. But how can I make such an assertion to my students? We seemingly want to use approximations everywhere but not think about how approximations have influenced the thinking of generations of scientists.
The basic underlying principles of any complicated aspect of nature can be obtained only if we make appropriate approximations. The evolution of science is often driven by measurement uncertainties, approximations, estimates, unphysical assumptions and often well thought out speculations. Einstein is clear about it when he said, “As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” Approximations help us in bridging the gap between what is not certain and what is reality. As practitioners of science, we know their importance. But are we sensitizing our students enough about it? I think it is not adequate just to “derive” a formula or a theory but it is also essential to discuss the underlying philosophy of the approximations we make. Otherwise, students may often think that the approximations do not make “sense”. They may not even appreciate the importance of acquiring the ability to make approximations.
When you are faced with a complex situation, the solution looks intractable. We need to break down the complexity into a simpler form. How can we do that without approximations? To make this point clear, let us look at some outstanding “approximations” that influenced the future of science.
For a hardened science practitioner, making approximations may sound unremarkable but not to the student. Simply knowing or being told that pi has an approximate value is of little relevance unless the student is made to appreciate how the idea of finding an approximation to pi actually led to new knowledge. Archimedes’s persistence in using the method of exhaustion to find a better approximation to Pi ultimately paved the way to a new area of mathematics called integral calculus. But often we fail to emphasize this connection leaving the student a chance to appreciate why approximations are important.
Since the time Copernicus pointed out that the Sun is the center of the solar system, everyone believed that planets moved in circular orbits at a constant speed. Kepler did not like this idea since it was not fitting the best available data of that time for planet Mar’s orbit. After many frustrating efforts, Kepler found that elongated circles or ellipses fitted the data nearly well. He knew that an ellipse is an approximation to a circle when the foci are brought closer. He simply kept the Sun at one of the foci of the ellipse. This led to Kepler’s first law. He, however, was not satisfied with the results leading him to postulate that the speed of the planet changed as it approached the Sun. This became his second law. His effort was an unthinkable scientific feat. Kepler’s work, published in the year 1609, is a historical example of how one can use limited data but make inspired guesses using appropriate approximations.
We need to provide students the appreciation of the unity of approximation and innovation. It was Edward Lorenz, an MIT professor, who proved that the weather cannot be predicted with any reasonable accuracy beyond about two weeks. His mathematical models, consisting of just three variables and three equations, indicated that there are formal predictability limits for certain deterministic systems. Lorenz found this accidentally. This brilliant insight that we have to accept approximate predictions when dealing with large dynamic systems, such as the atmosphere led to a new scientific field called the chaos theory. After relativity and quantum mechanics, the chaos theory is considered to be a third scientific revolution of the 20th century. Lorenz will be remembered for making us aware that weather forecasting will always remain approximate. The Kyoto Prize committee noted that Lorenz “has brought about one of the most dramatic changes in mankind’s view of nature since Sir Isaac Newton”.
Factorials are used very commonly in algebra, calculus, probability theory and number theory. Using successive multiplication, one can calculate the factorial of any non-negative integer n. But from a computing point of view, that is an utterly inefficient way of finding the factorial if n is large. Large factorials can easily be computed using approximations such as Stirling’s approximation. If you are not a mathematician, you may not be aware that Srinvasa Ramanujan’s approximation to factorial n is even more accurate than Stirling’s approximation as the value of n increases. Ramanujan is well-known for many other approximations in number theory and his approach to obtain these approximations is unparalleled in the history of mathematics since the time we began our efforts to approximate the value of pi about 4000 years ago.
Sometimes scientists abandon well respected old ideas and theories only to find themselves amidst new revelations. Perturbation theory is a great mathematical approach which helps us find approximate solutions to problems to which it may be impossible to find exact solutions. This theory has its origins in a concept called ‘linear approximation’ in use since 17th century in physics. Hooke’s law is a famous example of linear approximation. Let me explain it in simple words.
Hooke’s law, discovered by Robert Hook in 1660, states that “the force F a spring experiences is proportional to the distance x it is deformed from its natural length L”. Hooke’s law is expressed as F = -kx where k is a constant. There cannot be a simpler equation to describe the elastic behavior of a spring. However, it is clear that the spring cannot be compressed to zero thickness or it cannot be stretched to infinite lengths where it will break. Hooke’s law will simply fail at these extreme lengths. For a limited range of x, as long as the deformation is “small”, we see that Hooke’s law is a linear approximation and works quite effectively. Ohm’s law, equations for thermal expansion or pendulum movement all of which are linear approximations based on experimental observations. In the absence of a deeper appreciation of linear approximations, Raleigh and Schrodinger, could not have invented the perturbation theory without which there is no possibility of finding simple solutions to describe complex quantum systems.
When we want to describe reality, space and time remain as our fundamental conceptual aids. However, a recent approximation developed by scientists is expected to break this notion. Particles, spread all over the universe, interact constantly with each other and are the most basic events intrinsic to Nature. The laws that govern the particle interactions are described by the quantum field theory. The complexity of these equations, running into several thousands of terms, needed to capture the nature of elementary particles and their interactions is overwhelming. But all that is going to change dramatically with the discovery of a jewel like geometric object, called “amplituhedron” by Nima Arkani-Hamed of Institute of Advanced Studies, Princeton, N.J. Finding the volume of this amplituhedron, Arkani-Hamed’s team has demonstrated that the complex equations of quantum field theory can be reduced to an equivalent simple expression. With this advancement, the researchers claim, one can make the quantum field theory computations even on the back of an envelope without the need for advanced computers.
Mathematical description of gravity at the quantum scale is an uphill-task due to the possibility of sudden surprises involving inexplicable paradoxes and infinities. The physics world is now abuzz with the likelihood of developing a simple unified theory of quantum gravity (to combine the large and small-scale understanding of the universe) using similar geometrical approximations. This area of study is evolving and not all scientists are comfortable with the idea that we can throw away the notions of time and space and use only simple geometrical approximations. However, even in the 21st century, approximations continue to be a “hot” topic and draw fiery debates among scientists because they could shatter our rigid deep rooted beliefs of reality and present mother Nature in a more comprehensible and simple form.
Let us rejoice in the centuries old practice of “approximations”. Let us exhort our students to look for opportunities to approximate. If you can underline “approximations” as an important tool of scientific pursuit to be mastered and make students pay serious attention and not simply stare over them, we shall have accomplished a valuable and significant task as professors. Einstein once said about God, ““I want to know His thoughts. The rest is just details.” For students it would be more fascinating to know how scientists worked toward developing a scientific idea using approximations than knowing the idea itself. But for this to happen, we have to get away from being routine teachers and instead work harder to become effective teachers, a species fast becoming rarer than the hen’s teeth.
Approximations are the lifeline to solve the most complicated aspects of nature. Some professors do emphasize the philosophical and historical aspects of approximations in their lectures. Why not every professor? Who knows if one of our students enthused by our emphasis on the importance of approximations would turn out to be a future Ramanujan or Kepler or Schrodinger? Don’t you agree?
PS: “Telling lies to describe truth” in my title is used only to catch your attention. I do not certainly mean approximations are “lies”.